Novel collective excitations in a hot scalar field theory
aa r X i v : . [ h e p - ph ] J a n TUM-HEP-914/13
Novel collective excitations in a hot scalar field theory
Marco Drewes
Physik Department T70, Technische Universit¨at M¨unchen,James Franck Straße 1, D-85748 Garching, Germany
Abstract
We study the spectrum of quasiparticles in a scalar quantum field theory at high temperature.Our results indicate the existence of novel quasiparticles with purely collective origin at lowmomenta for some choices of the masses and coupling. Scalar fields play a prominent rolein many models of cosmology, and their collective excitations could be relevant for transportphenomena in the early universe.
Quantum field theory provides the most fundamental description of matter and radiation we knowand solves the apparent “wave particle dualism” in a consistent way. With quantised fields beingthe fundamental building blocks of nature, the elementary excitations of these fields in weaklycoupled systems propagate like particles. Curiously, there are not only elementary particles; in amedium the collective excitations of many elementary quanta often effectively behave as if theywere particles themselves.Often one is not interested in the fate of individual particles, but mostly in transport of energyor charges within a system. Transport phenomena can be studied in a thermodynamic descriptionin terms of a density matrix ̺ . The propagator in this effective thermodynamic description canhave a rather different structure than in vacuum. This reflects the fact that propagating particlesare affected by the medium. In weakly coupled systems this effect can often be parametrised byinterpreting the poles of the propagator as quasiparticles with modified properties. For instance,the dispersion relations (or “bands”) of electrons in a solid state can be very different from that invacuum. Also the effective charge is screened in a medium. In addition to the screened elementaryparticles there can be new types of quasiparticles that have no analogue in vacuum. These canbe interpreted as quantised collective excitations of the background medium. For instance, in asolid state the lattice vibrations, phonons, behave like quasiparticles. The existence of collectiveexcitations is also well-known from relativistic quantum field theory. In gauge theories withcoupling α ≪ T there are fermionic excitationswith soft momenta p ∼ αT [1–5] and ultrasoft momenta p ∼ α T [6, 7] which have no analoguein vacuum. These are often referred to as holes or plasminos . Collective fermionic excitations havealso been found in models with Yukawa interactions [8–10]. Also longitudinal gauge bosons appearat finite temperature with a dispersion relation that differs from the transverse components.In this work we find evidence that collective excitations can also exist in purely scalar fieldtheories. The existence of collective propagating modes in principle is expected; in particularhydrodynamic modes, such as sound waves, should appear in the spectrum of any field theory.However, to the best of our knowledge, quasiparticles beyond the hydrodynamic regime have notbeen described explicitly in the context of purely scalar field theories. On one hand their existencecan simply be viewed as an interesting property of the field theory. On the other hand, currentexperimental evidence [11, 12] suggests that there is at least one scalar field in nature, the Higgsfield. Furthermore, many models of cosmology involve additional scalar fields, such as axions,the inflaton, dilaton, moduli fields or Affleck-Dine fields. Since the universe was exposed to very1igh temperatures during the early stages of its history, the spectrum of scalar quasiparticles mayhave affected transport phenomena in the early universe. We consider a simple model of two scalar fields described by the Lagrangian L = 12 ∂ µ φ∂ µ φ − m φ φ + 12 ∂ µ χ∂ µ χ − m χ χ − gφχ . (1)We choose this Lagrangian for illustrative purposes, as it describes the (probably) simplest scalarmodel in which the additional collective excitations we found appear. We expect that similarbehaviour can be found in more realistic models where the structure of the self-energies is similar. Following the approach of [13–15] we study the system in terms of real time correlation func-tions. This approach has been applied to scalar fields in different situations [16–31] relevant forcosmology. We use the notation of [20]. The expectation values or one-point functions h φ ( x ) i and h χ ( x ) i play the role of the “classical field”. The average h . . . i is defined in the usual way as hAi = Tr( ̺ A ), where ̺ is the density matrix of the thermodynamic ensemble. It includes theusual quantum average as well as a statistical average over initial conditions. We will in the fol-lowing assume that all degrees of freedom are in thermal equilibrium and set h φ ( x ) i = h χ ( x ) i = 0.Quasiparticle properties are encoded in the propagator or two-point function. We can define twoindependent two-point functions for φ ,∆ − ( x , x ) = i ( h φ ( x ) φ ( x ) i − h φ ( x ) φ ( x ) i ) (2)∆ + ( x , x ) = 12 ( h φ ( x ) φ ( x ) i + h φ ( x ) φ ( x ) i ) , (3)and analogously for χ . ∆ − is called the spectral function . It encodes the properties of quasiparti-cles and is the main quantity of interest in this work. ∆ + is called the statistical propagator andcharacterises the occupation numbers of different modes. Out of thermal equilibrium, ∆ − ( x , x )and ∆ + ( x , x ) would be two independent functions, and each of them would depend on x and x individually. Thermal equilibrium is homogeneous, isotropic and time translation invariant,hence the correlation functions can only depend on the relative coordinate x − x . This allowsto define the Fourier transform ρ p ( p ) = − i Z d ( x − x ) e ip ( t − t ) e − i p ( x − x ) ∆ − ( x − x ) . (4) Note that the energy functional obtained from (1) is not bound from below. For the purpose of illustrating theappearance of collective scalar quasiparticles we will ignore this issue here and consider small excitations aroundthe local minimum at φ = χ = 0, assuming that (1) is embedded into a bigger framework that stabilises the groundstate. Furthermore, in thermal equilibrium ∆ − and ∆ + are not independent, but related by the Kubo-Martin-Schwinger relation, which for their Fourier transforms reads ∆ + p ( p ) = f B ( p )2 ρ p ( p ). Here f B is the Bose-Einstein distribution. This is the quantum field theoretical version of the detailed balance relation.
2t can be expressed as [20] ρ p ( p ) = − R p ( p ) + 2 p ǫ ( p − m − p − ReΠ R p ( p )) + (ImΠ R p ( p ) + p ǫ ) . (5)Here Π R p ( p ) is the Fourier transform of the usual retarded self-energy, in this caseΠ Rφ ( x , x ) = g θ ( t − t ) (cid:16) χ ( x ) χ ( x ) χ ( x ) χ ( x ) − χ ( x ) χ ( x ) χ ( x ) χ ( x ) (cid:17) , (6)and analogous for χ . In (5) we have not specified whether we refer to φ or χ ; both spectraldensities formally have the same shape except for the replacement m → m φ or m → m χ and theinsertion of the corresponding self-energy. The pole structure of ρ p ( p ) in the complex p planedetermines the spectrum of quasiparticles. In vacuum there would be only one pole for positive p at p = ω p ≡ ( p + m ) / , where m is the renormalised mass. At T > i . We refer to the pole that converges to ω p in the limit T → screened one-particle state , and to all other poles as purely collective excitations.Π R p ( p ) can be expressed as the sum of a vacuum contribution and a temperature dependentmedium correction. The real part of the vacuum contribution contains the usual UV divergencethat also appears in vacuum, the temperature dependent part is UV-finite. It is common toimpose renormalisation conditions at T = 0 to absorb the divergence and define the physicalmass [19, 20]. We will in the following simply interpret m φ and m χ as physical masses in vacuumafter renormalisation and ReΠ R p ( p ) as the remaining finite piece. Let ˆΩ i p be a pole of ρ p ( p )with Ω i p ≡ Re ˆΩ i p and Γ i p ≡ i p . Ω i p and Γ i p are temperature dependent because Π R p ( ω )depends on T . In weakly coupled theories one usually observes the hierarchyΓ i p ≪ Ω i p . (7)Due to (7) we can interpret Ω i p as a quasiparticle dispersion relation (or “thermal mass shell”)and Γ i p as its thermal width (or damping rate). Near poles that fulfil (7) the spectral density canbe approximated by ρ BW p ( p ) (cid:12)(cid:12) p ≃ Ω i p ≃ X i Z i p p Γ i p (cid:0) p − (Ω i p ) (cid:1) + (cid:0) p Γ i p (cid:1) + ρ cont p ( p ) (8)Here the residue and width are given by Z i p = " − i p ∂ ReΠ R p ( p ) ∂p − p =Ω i p , Γ i p = −Z i p ImΠ R p (Ω i p )2Ω i p . (9)In the zero-width limit the it reads ρ p ( p ) = X i Z i p π sign( p ) δ (cid:0) p − (Ω i p ) (cid:1) + ρ cont p ( p ) , (10) Formally we should use different symbols for the mass parameter appearing in (1) and full self-energy beforerenormalisation on one hand and the physical mass and finite part of ReΠ R on the other. However, the former donot appear anywhere in the following calculation. We refer to any pole of a propagator that fulfils (7) as quasiparticle, may it be a screened one-particle state ora collective excitation, and regardless of its spin. ) b ) Figure 1:
Diagrams contributing to the self-energies for φ , a ) , and χ , b ) , at on-loop order. Solid linesrepresent φ -propagators, dashed lines χ -propagators. which can be compared to the free spectral density ρ free p ( p ) = 2 π sign( p ) δ ( p − ω p ) . (11)The dispersion relation in (10) is essentially fixed by ReΠ R p ( p ) via the condition p − p − m − ReΠ R p ( p ) = 0 . (12)For this reason the real and imaginary part of the retarded self-energy are often referred to asthe dispersive self-energy and dissipative self-energy, respectively.The dispersion relations Ω i p can have a complicated p -dependence. In limited momentumregimes they can often be approximated by momentum independent “thermal masses”. For hardmodes p ∼ T it is common to define the asymptotic mass M , which depends on T but noton p , by fitting the approximation ( p + M ) / to the full dispersion relation in the regime p & T . This approximation is commonly used in transport equations because most particles in aplasma in thermal equilibrium have momenta p ∼ T . In this work we are interested in collectiveexcitations. These usually appear in the momentum regime p ≪ T , where the energy related tointer-particle forces can be comparable to their kinetic energy or larger. Therefore we cannot usethis approximation. For the Lagrangian (1) the leading order contribution to the φ -self-energy comes from the diagramshown in figure 1a). Using finite temperature Feynman rules [32], the imaginary part of thisdiagram can be calculated fromImΠ Rφ p ( p ) = − g Z d k (2 π ) (1 + f B ( k ) + f B ( p − k )) ρ χ k ( k ) ρ χ p − k ( p − k ) , (13)where the subscript indicates the self-energy or spectral density of which field we mean. At one-loop level the integral (13) is to be evaluated with free spectral densities (11). This correspondsto using free thermal propagators in the loop. The result is − ImΠ
Rφ, p = g π | p | " − θ ( − p ) p + T log (cid:20) f B ( p − ω + φ ) f B ( − ω − φ ) f B ( − ω + φ ) f B ( p − ω − φ ) (cid:21)! (14)+ θ (cid:0) p − (2 m χ ) (cid:1) T log (cid:20) f B ( p − ω + φ ) f B ( − ω − φ ) f B ( − ω + φ ) f B ( p − ω − φ ) (cid:21) .0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 - - - - p (cid:144) m Φ R e P ï R H p L Figure 2:
The real part of the retarded φ -self-energy Re ˆΠ Rφ p ( p ) ≡ ReΠ Rφ p ( p ) /g for m χ = m φ / , T = 333 m φ and | p | = m φ / as a function of p . The function is positive for p ≫ m φ . with the Bose-Einstein distribution f B ( p ) ≡ ( e p /T − − and ω ± φ = p ± sign( p ) | p | s − (2 m χ ) p (15)The real and imaginary part of the self-energy are related by the Kramers-Kronig relations, whichallow to rewrite (12) as p − p − m φ − P Z dωπ ImΠ Rφ p ( ω ) ω − p = 0 . (16)In (16) we neglect the T = 0 part of (14), as this contribution has been absorbed into m φ already.We now choose a set of parameters g = m χ = m φ / T = 333 m φ and | p | = m φ /
10. Thereal and imaginary part of the self-energy for this choice are shown in figure 2 and 3. The mostprominent features in ReΠ Rφ p ( p ) are two spikes, which appear near p = p + (2 m χ ) becausethe zero in the denominator of the principal value term in (16) passes the kinematic thresholdsof (14) in the numerator. Another remarkable feature is that the finite temperature correction isnegative in that p -region. This leads to a negative thermal mass correction, and it is preciselythe reason why we find more than one solution to (16). The negative mass shift seems unusualfrom a particle physics viewpoint, where one is used to positive “thermal masses”. Note, however,that a frequency Ω p that is smaller than in vacuum for a given p in optics corresponds to a indexof refraction greater than one, i.e. the normal behaviour. The left hand side of (16) is plotted infigure 4. There are three solutions to (16), which we label by Ω a p , Ω b p and Ω c p . One of them isat p ≃ ω p and can be interpreted as dressed one-particle state. The resulting spectral density ρ φ, p ( p ) is shown in figure 5.The first peak from the right lies at Ω a p ≃ ω p . It is the dressed one-particle state. It isinteresting that this pole lies so close to the vacuum mass shell. This implies that thermal mass5 .01 0.05 0.10 0.50 1.00 5.00 10.000.0010.010.1110 p (cid:144) m Φ - I m P ï R H p L Figure 3:
The imaginary part of the retarded φ -self-energy Im ˆΠ Rφ p ( p ) ≡ ImΠ Rφ p ( p ) /g for m χ = m φ / , T = 333 m φ and | p | = m φ / as a function of p . The dotted line is the T = 0 contribution. - - p (cid:144) m Φ p - p ® - m Φ - R e P R H p L Figure 4:
The left hand side of (16) for g = m χ = m φ / , T = 333 m φ and | p | = m φ / as a functionof p . .0 0.2 0.4 0.6 0.8 1.0 1.2 1.4246810 p (cid:144) m Φ Ρ p H p L (cid:144) m Φ Figure 5:
Solid red line: The spectral density ρ φ p ( p ) from (5) with g = m χ = m φ / , T = 333 m φ and | p | = m φ / as a function of p . The isolated feature near p ≃ . m φ represents a δ -function,which we interpret as a novel scalar plasmon of purely collective origin, the luon . Another interestingfeature is the continuous contribution for p < , which disappears in the limit T → . Dotted blackline: The Breit-Wigner approximation (8) in the vicinity of the screened particle pole. corrections are negligible for low momentum modes in the model described by (1), which hasalready been observed in [33]. This is also confirmed by the analysis in appendix A. Naively onemight have expected that the thermal correction dominates over the vacuum mass for gT ≫ m φ .In spite of the small thermal mass corrections, thermal effects strongly dominate the width: Γ a p is about ∼ leading to a significant broadeningof the peak. With Γ a p / Ω a p ≃ . ρ p ( p ) by the Breit-Wigner function (8) near the peak works at reasonableaccuracy, see figure 5.The remaining two poles at Ω b p ≃ . m φ and Ω c p ≃ . m φ are due to the negative spikein ReΠ R p ( p ). They lie very close to each other; their separation is considerably smaller thanΓ a p . They do, however, not merge into a single resonance because Ω c p has vanishing width in theone-loop approximation (14). This is because (14) vanishes below the two-particle threshold, asthe only processes contributing to ImΠ Rφ, p ( p ) at one-loop level on-shell are decays and inversedecays φ ↔ χχ . Therefore Ω c p is associated with a stable quasiparticle in this approximation. Werefer to this novel, purely collective l ow moment u m scalar plasm on as luon .The solution Ω b p , on the other hand, is broadened and absorbed by ρ cont φ, p ( p ) near the threshold,where ImΠ Rφ, p ( p ) is relatively large. This implies that there is no propagating quasiparticleassociated with Ω b p , as the lifetime of this resonance would be so short that its mean free pathin the plasma is shorter than its de Broglie wavelength. The Breit-Wigner approximation isnot valid near Ω b p because (14) is a steep function in the regime near the two-particle threshold.Hence, we keep the full p -dependence of Π Rφ, p ( p ) in (5) in all plots. This is also obvious from the In figure 3 it can be seen that thermal effects strongly dominate ImΠ
Rφ, p ( p ). c p in the very same approximation(14) has infinite lifetime. It shows that the spectrum of quasiparticles is very sensitive to thethreshold behaviour of ImΠ R p ( p ), which may be affected by higher order corrections. We have studied the spectrum of quasiparticles in a hot scalar field theory. We calculated thespectral density ρ φ p ( p ) at one-loop level. For our choice of parameters we find that ρ φ p ( p )has three poles for some momenta in the regime p < m φ . One of them can clearly be identifiedwith the dressed one-particle state. It shows significant thermal broadening, but only a negligiblethermal mass shift. It can be described by a Breit-Wigner approximation. Of the remaining twopoles, one can be interpreted as a new quasiparticle while the other one is broad and cannot beinterpreted as a quasiparticle. Both have very similar energies and lie in the threshold region.Therefore our results are rather sensitive to loop corrections in this region. As a first cross-check for the validity of our result, we evaluate the sum rule Z dp π p ρ p ( p ) = 1 , (17)which directly follows from the commutation relations [ φ ( x ) , ˙ φ ( x )] | t = t = iδ ( x − x ) for ascalar field. Numerical evaluation of (17) with (5) yields 0 .
996 for our parameter choice, i.e. goodagreement. In order to estimate the importance of the different features in ρ φ p ( p ) for transport,it is instructive to evaluate the contribution to (17) from different p -regions separately. Theregion p < c p contributes about 6%. This suggests thatthe collective excitations generally do not contribute much to phase space integrals. The exchangeof luons will typically affect transport in the plasma even less than in the case of fermionic holesin gauge theories: both types of collective excitations only exist for small momenta p ≪ T (while most particles in a relativistic plasma have momenta p ∼ T ), but the fermionic holes haveresidues of order one for soft momenta. The region around Ω b p contributes about 8% to the sumrule (17). The main contribution of 72% comes from the region around the screened one-particlepole at Ω a p . This suggests that transport in the low momentum region predominantly happensvia the exchange of these short lived resonances. Finally, there is a contribution of 13% from thecontinuum contribution at p > Ω a p .In spite of their small residue luons could make a relevant contribution to transport whenthe processes involving the dressed particles are kinematically forbidden. A situation of thiskind has been studied in detail in [33] for the relaxation of the zero-mode of a massive scalarfield in a plasma of scalars and fermions with gauge interactions at temperatures larger thanthe scalar’s mass. The scalar may be identified with the inflaton, an axion, moduli-field or anorder parameter during a phase transition. The relaxation rate is sensitive to the quasiparticlespectrum at momenta k ≪ T because the momenta of the decay products are of the order of thescalar mass, and not of the order of the temperature. In some temperature regimes the processesinvolving dressed particles can be kinematically blocked by large thermal masses of the decayproducts. It was found in [33] that in this situation it is crucial to take into account the full8uasiparticle spectrum at low momentum. While this raises hope that there may be physicalsystems in which the exchange of luons is relevant, it also forces us to question whether the use ofthe one loop result (14) is justified. The new scalar plasmons lie in the threshold region, wherethe shape of the self-energy and spectral density are usually sensitive to corrections of higherorder in the loop expansion. Some of these corrections can be taken into account by using full χ -propagators in the loop in figure 1a). This is expected to have two different effects.On one hand the full χ -propagators include a finite width due to the imaginary part ofthe thermal χ -self-energies. The expression (14) for ImΠ Rφ p ( p ) obtained using free thermal χ -propagators vanishes below the two-particle threshold p = p + (2 m φ ) , see figure 3. It is clearthat ImΠ Rφ p ( p ) is non-vanishing along the entire p axis once contributions of higher order in theloop expansion are taken into account [22, 33], which can be interpreted as damping by scatteringsand lead to a Γ c p = 0. This width smears out the kinematic thresholds [22, 33] and leads to anon-vanishing ImΠ Rφ p ( p ) at p < p + (2 m φ ) . Due to the very small splitting Ω b p − Ω c p it seemspossible that the peaks in ρ p ( p ) at Ω b p and Ω c p merge and effectively form a single resonance whenthese processes are taken into account. Whether or not this resonance is sufficiently long-livedto be interpreted as a propagating quasiparticle depends on the size of the corrections. Anothereffect of smoothing out the kinematic thresholds is that the spike in ReΠ Rφ, p ( p ) is less sharp.This is because the spike appears where the pole in the denominator of the integral in (16) passesthe kinematic threshold. Also this tends to have the effect of broadening the resonances. Thesize of Γ c p cannot be determined without a proper calculation. It is known that in some cases theeffects of multiple scatterings entirely overcome the suppression of ImΠ Rφ, p ( p ) in the region thatis kinematically forbidden at one-loop level [37]. This would possibly eliminate the luon fromthe quasiparticle spectrum. However, there are also situations in which the suppression remainswhen higher order corrections are taken into account [33].The other important effect of using dressed χ -propagators are the modifications to the χ -dispersion relations due to the real part of the χ -self-energy. Thermal mass corrections can shiftthe two-particle threshold in ImΠ Rφ, p ( p ), and hence the position of the spike in ReΠ Rφ, p ( p ),to larger values of p . This is crucial as ReΠ Rφ, p ( p ) can compete with p − p − m φ in (16)only because of the spike, and it can do so only if the spike lies in the region p . m φ , where p − p − m φ is small. If thermal masses would shift the spike to the region p ≫ m φ , thenReΠ Rφ, p ( p ) < p − p − m φ for all p , and there is only one solution to (16). The analysis in thefollowing section and appendix A suggests that this is not the case for our choice of parameters. Also in scalar QED the dissipation rate is sensitive to the spectrum in the infrared. At one-loop level it isformally of order α [31] due to processes φ ↔ φγ : While in vacuum the decay φ → φγ is kinematically forbidden,it formally leads to a finite dissipation rate at T = 0 because the vanishing phase space volume is compensated bythe infinite occupation number for photons with energy zero. This unphysical behaviour arises because the thermalphoton mass has been neglected, which regularises the infinite occupation number leads to the kinematic blocking.Once the thermal photon mass is included the on-shell decay of a scalar particle into itself and a photon cannotcontribute to the dissipation any more. If collective scalar plasmons are present in the spectrum, then the decayof a screened scalar particle into a photon and such a plasmon could still give a non-zero contribute (along withdifferent scattering processes obtained from other cuts through the self-energies [22, 33–36]). The parameters used in the examples given in [33] were deliberately chosen to avoid the issues we discuss inthe following. In addition, there may be vertex corrections due to “ladder diagrams” of the same order, which we do notdiscuss here. We assume that they simply lead to a change in the effective coupling constant or are of higher order.For Yukawa type vertices it has been argued that this assumption is applicable to diagrams of this topology [8],but for gauge interactions it is not [37]. .2 The χ propagator at one loop level To estimate the thermal corrections to χ -properties we evaluate the χ -self energy Π Rχ p ( p ) infigure 1b) with free thermal propagators in the loop, i.e. using free spectral densities (11). Theimaginary part is given byIm Π Rχ, k ( k ) = − g Z d q (2 π ) (1 + f B ( q ) + f B ( k − q )) ρ φ q ( q ) ρ χ k − q ( k − q ) (18)and can again be computed analytically, − ImΠ
Rχ, k = g π | k | " − θ ( − k ) k (19)+ (cid:16) θ (cid:0) k − ( m φ + m χ ) (cid:1) − θ (cid:0) − k + ( m φ − m χ ) (cid:1)(cid:17) T log (cid:20) f B ( k − ω + χ ) f B ( − ω − χ ) f B ( − ω + χ ) f B ( k − ω − χ ) (cid:21) with ω ± χ = 12 k h k ( m φ − m χ + k ) ± | k | q m χ + ( m φ − k ) − m χ ( m φ + k ) i (20)Due to the different processes, there are two thresholds. This leads to the appearance of a double-spike in the real part in figure 6, which we again calculate using the Kramers-Kronig relations.Since both thresholds involve the heavier mass m φ , the spikes lie in a region where they cannotcompete with k − k − m χ , hence there is no collective excitation. We now use this result todiscuss the spectrum of χ -quasiparticles.We first discuss the χ -spectral function for momenta k ∼ m φ . The real and imaginary parts ofthe χ -self energy Π Rχ p ( p ) for | k | = m φ are shown in figures 6 and 7. Figure 8 shows the χ -spectraldensity, obtained by inserting the above results for Π Rχ, k ( k ) into (5). The quasiparticle spectrumcan be studied by solving the equation k − k − m χ − Re Π Rχ k ( k ) = 0 . (21)For the parameters we considered it consists of a single peak, which is located slightly belowthe free particle energy ( k + m χ ) / in vacuum, i.e. there is a small negative thermal massshift. This mass shift, however, does not endanger the existence of the new scalar plasmon -if the diagram 1a) is evaluated with smaller m χ , then the spike in figure 2 moves to the leftand remains in the regime where it can overcome p − p − m φ in (16). As for the screened φ -particle, we observe a considerable thermal broadening. The sumrule (17) is fulfilled. We canalso calculate the χ -spectral function for hard modes k ∼ T , which is shown in figure 9. Asexpected, it consists of one sharply defined quasiparticle. The screening leads to a small positivemass shift in this hard momentum region. This confirms that it is, up to finite width corrections,a reasonable approximation to use free thermal χ -propagators when calculating the φ -self-energyfor loop momenta k ∼ m φ and k ∼ T . This conclusion agrees with what we find in appendixA. Note that our analysis here goes beyond the hard thermal loop approximation, as (19) wasobtained analytically without assumptions about the relative size of external and loop momenta.The situation becomes more complicated when looking at χ -modes with momenta k < m φ ,which also contribute to the χ -self energy 1a). Due to the smallness of m χ , it turns out that(21) evaluated with (19) has no solutions at all for very small momenta. One way to interpret10 .5 1.0 1.5 2.0 - - - p (cid:144) m Φ R e P ï R H p L Figure 6:
The real part of the retarded χ -self-energy Re ˆ Π Rχ p ( p ) ≡ Re Π Rχ p ( p ) /g for g = m χ = m φ / , T = 333 m φ and | p | = m φ as a function of p . this is that there are no χ -quasiparticles with very small momenta, i.e. there is a minimumwave number below which the damping by the plasma is so strong that there are no plane wavesthat propagate for at least one oscillation. An interpretation of this is that χ -particles travellingthrough the medium with very small momenta are “halted” by interactions with the backgroundmedium, which formally reflects in the “melting” of the quasiparticle peak for very soft modes athigh T . If this is what physically happens, it could have considerable effect on the φ -self-energy.The kinematic threshold in (14), which is responsible for the appearance of the spikes in figure2, is caused by the energy conserving δ -functions in (11). As long as all widths remain narrow,thermal corrections encoded in (10) will only move the mass shells and threshold. But if thepeaks for soft modes indeed “melt”, then (11) or (10) by no means are a good approximation forthe modes k , p − k ≪ m φ in the loop. This could smear out the threshold in (14) and hence thespike in figure 2, which might eliminate the luon-solutions of (16) from the spectrum. However,within our numerical precision the χ -spectra for | k | ≪ m φ obtained from (19) fail to satisfy thesum rule (17) with an error of order one. This indicates that our treatment of these modes isinsufficient, and we can in fact not make any reliable statement on the χ -quasiparticle spectrumat k ≪ m φ . The problem is that we obtain the quasiparticle spectra from the diagrams in figure 1, the eval-uation of which already requires knowledge of the full propagators (and thus the quasiparticlespectrum) in the loop. In principle corrections to Π
Rφ, p ( p ) are of higher order in the loop ex-pansion and involve more vertices, hence one could expect them to be suppressed by powers of g/m φ . However, naive loop counting cannot always be applied in thermal field theory at hightemperature because large occupation numbers can compensate for the suppression by additionalvertices. Formally this happens because the Bose-Einstein distribution in the thermal propa-11 .0 2.01.55.6.7.8.9.10. p (cid:144) m Φ - I m P ï R H p L Figure 7:
The imaginary part of the retarded χ -self-energy Im ˆ Π Rχ p ( p ) ≡ Im Π Rχ p ( p ) /g for g = m χ = m φ / , T = 333 m φ and | p | = m φ as a function of p . p (cid:144) m Φ Ρ p H p L (cid:144) m Φ Figure 8:
The spectral density ρ χ p ( p ) with g = m χ = m φ / , T = 333 m φ and | p | = m φ as a functionof p .
100 200 300 400 50001. ´ - ´ - ´ - ´ - ´ - ´ - p (cid:144) m Φ Ρ p H p L (cid:144) m Φ Figure 9:
The spectral density ρ χ p ( p ) for χ with g = m χ = m φ / , T = 333 m φ and | p | = T as afunction of p . gators can bring powers of the coupling into the denominator, so that the loop expansion neednot be identical with the perturbative expansion. This problem is sometimes referred to as a“breakdown of perturbation theory” [38].It can in some cases be fixed by using resummed propagators and vertices in the loops, see e.g.[39]. Formally the resummed perturbation theory is nothing but a consistent perturbation theory,i.e. a systematic expansion in the coupling constant. A scheme that is commonly used in gaugetheories and for Yukawa interactions is the hard thermal loop (HTL) resummation technique[40–42]. The HTL resummation is a consistent expansion in a coupling constant α due to aseparation of scales for α ≪
1. Thermal corrections to the dispersion relations of order ∼ αT arerelevant for soft external momenta p . αT . In contrast to that, the loop integral in the regime T ≫ m φ , m χ is usually dominated by hard loop momenta k ∼ T , for which thermal correctionsare negligible. This justifies to use free thermal propagators inside the loop when calculatingradiative corrections to propagators with soft external momenta. The corrected propagatorsobtained this way can then be used in all further calculations.Unfortunately this strategy cannot be applied in the model (1) because the self-energy diagramin figure 1a) for a particle with momentum p ≪ T is, in contrast to gauge theories, not dominatedby hard momenta ∼ T inside the loop. The reason is that the self-energies contain only scalarpropagators. In comparison to, for instance, the fermion propagators appearing in the photonself-energy, these give less powers of momenta. The need to use resummed propagators in loops can also be understood from simple kinematic arguments: Ifone uses the free thermal propagators given by (11) in the loop while assigning thermal masses to the externalparticle, then two propagators of the same particle with the same momentum meeting at a vertex would havedifferent effective masses, which seems clearly unphysical. Loop corrections have recently been studied beyond the HTL scheme in different contexts [22, 33, 36, 37, 43–46], but these results from Yukawa and gauge theories cannot be applied to our model. Resummations in scalarfield theory have been studied in detail for the λφ -interaction, see e.g. [32, 33, 39, 47–50]. For that interactionconsiderable simplification arises from the fact that the leading order correction is a momentum-independentthermal mass of order √ λT from a local diagram. This is not the case in the model (1). p be the external momentum of Π Rφ, p ( p ) and k the momentum inside the loop. If p ishard, i.e. comparable to T or larger, then the thermal correction to the dispersion relation issmall compared p . The same argument justifies the use of free propagators in the part of theloop integration volume where k is hard. This allows to treat the effect of hard loops by meansof resummed perturbation theory. The uncertainty in our calculations of Π Rφ, p ( p ) arises fromthe region where p and k are both small compared to T and m φ , i.e. from the interaction ofthe low momentum modes with each other. Similar uncertainties also exist in gauge theories,but there the loop integrals are usually dominated by the region k ∼ T , hence they only affecta sub-dominant contribution to the self-energy. In the scalar model under consideration here, incontrast, the integration region k ≪ T considerably contributes to Π Rφ, p ( p ). A fully consistenttreatment of this issue goes beyond the scope of the current work. Our simple estimate inappendix A suggests that thermal mass corrections to low momentum modes remain sufficientlysmall in the parameter region we consider. We postpone a more precise study to future work. We have studied the spectrum of quasiparticles in a scalar model at high temperature at one looplevel. Our results indicate the existence of novel quasiparticles related to collective excitationsin addition to the screened one-particle state for momenta p . m ≪ T , which we refer to as luons . The existence of purely collective excitations is in principle expected in any field theoryat finite temperature at least for very long wavelength, where hydrodynamic modes contributeto transport. In gauge and Yukawa theories collective excitations are known to exist for softmomenta, i.e. much above regime where hydrodynamic excitations exist. To the best of ourknowledge, this phenomenon has not been described in a purely scalar field theory to date.The new resonances lie in the region of a kinematic threshold, hence their properties maybe sensitive to higher order corrections. We postpone the technically difficult treatment of theseto future work. If we assume that they do not considerably modify the luons or even eliminatethem from the spectrum, then it is instructive to pose the question how general their appearanceis. Mathematically they arise because the dispersive self-energy exhibits a prominent negativespike that originates from the existence of a kinematic threshold in the dissipative self-energy atone-loop level. It is related to the steep rise of (14) in the threshold region. This suggests thatluon-type plasmons can generally occur where the dissipative self-energy is a steep function of p , such as near kinematic thresholds. Hence, similar effects may also be expected in other, morerealistic systems.However, there are two generic properties of quasiparticle poles of this kind. First, since theyoccur due to a sharp feature or spike, they usually come with a suppressing residue factor Z i p dueto the steepness of the dispersive self-energy in such a feature, see (9). That means that theircontribution to phase space integrals is rather small, and the exchange of these quasiparticles willnot contribute significantly to transport unless other channels are kinematically blocked. Suchsituations have been studied in cosmology [22, 25, 33, 37, 51, 52]. Second, when their appearanceis related to a kinematic threshold, they tend to lie in the threshold region, where changes in theshape of the dissipative self-energy due to higher order corrections can have a strong effect onquasiparticle properties.Finally, we would like to add that the collective excitations discussed here are absent in themost studied scalar model, the λφ -interaction. The reason is that the dispersive self-energy in14he λφ -model receives a momentum-independent contribution of order λT from a local diagram(“tadpole”). Any negative contribution to it, which could lead to additional poles in the spectraldensity, is of higher order in λ . This makes even the long wavelength modes too heavy forcollective excitations of the type discussed here to occur. Acknowledgements - I am grateful to Mikhail Shaposhnikov, Dietrich B¨odeker and MiguelEscobedo for helpful comments. I would also like to thank Jin U Kang for pointing out that theresult (19) can be written in a very compact form. This work was supported by the GottfriedWilhelm Leibniz program of the Deutsche Forschungsgemeinschaft.
A Thermal mass and expansion parameter
As discussed in section 3.3 it is not immediately clear that the loop expansion provides a system-atic approximation scheme for small momenta in the model (1). We formulate an effective fieldtheory for the low momentum modes, using the method of dimensional reduction [53–57]. Thisprocedure is commonly applied to static problems and in principle not suitable to treat dynamicquantities, such as the propagator we are interested here. Hence, we do not use it for a systematictreatment of the problem, but only to identify the relevant dimensionless expansion parameter.Correlation functions for quantum fields in four dimensional Minkowski space at high tem-perature can be obtained by analytic continuation of correlators that have been calculated in atheory with three real spatial dimensions x and an imaginary time dimension iτ on the compactinterval from 0 to − iβ , with β = 1 /T [58]. This leads to the Euclidean action S E = Z β dτ Z d x L E (22)with L E = 12 ( ∂ µ ) φ + 12 m φ, φ + 12 ( ∂ µ χ ) + 12 m χ, χ + gφχ . (23)Note that in this section m φ, and m χ, are to be understood as bare masses. In this imaginarytime formalism the Fourier decomposition in energy is a discrete sum, φ ( x , τ ) = ∞ X n = −∞ φ n ( x ) e iω n τ = φ ( x ) + ∞ X n =1 (cid:0) φ n ( x ) e iω n τ + φ ∗ n ( x ) e − iω n τ (cid:1) , (24) χ ( x , τ ) = ∞ X n = −∞ χ n ( x ) e iω n τ = χ ( x ) + ∞ X n =1 (cid:0) χ n ( x ) e iω n τ + χ ∗ n ( x ) e − iω n τ (cid:1) , (25)where we have used the reality of the fields in the second equality. Here ω n ≡ πT n , where n is an integer. Formally the four-dimensional theory with two fields φ ( x , τ ), χ ( x , τ ) is equivalentto an theory in three dimensional Euclidean space with infinitely many fields φ n ( x ), χ n ( x ). Athigh temperature modes with n > p ≪ πT can be constructed by integrating them out. The resulting effective Lagrangian is only valid forcorrelation functions with spatial separation ≪ /T . Moreover, there is no more dependenceon time, so one is restricted to static phenomena. We inserting (24) and (25) into (22) we canperform the τ -integral analytically and write S E as a spatial integral over x . Since we are only15nterested in the light modes φ and χ we drop all terms that do not contain these from S E . Inorder to disentangle φ n and φ ∗ n we introduce new field variables A n = r T Re φ n , B n = r T Im φ n , C n = r T Re χ n , D n = r T Im χ n . (26)We also define ˜ φ ≡ φ √ T , ˜ χ ≡ χ √ T , ˜ g ≡ g √ T . (27)With these new variables we can write S E = Z d x " ˜ L + ∞ X n =1 L n + L int (28)with ˜ L = 12 (cid:16) ( ∂ i ˜ φ ) + m φ, ˜ φ + ( ∂ i ˜ χ ) + m χ, ˜ χ (cid:17) + ˜ g ˜ φ ˜ χ , (29) L n = 12 (cid:16) ( ∂ i A n ) + ( m φ, + ω n ) A n + ( ∂ i B n ) + ( m φ, + ω n ) B n +( ∂ i C n ) + ( m χ, + ω n ) C n + ( ∂ i D n ) + ( m χ, + ω n ) D n (cid:17) , (30) L int = ˜ g ∞ X n =1 h χ ( A n C n + B n D n ) + ˜ φ (cid:0) C n + D n (cid:1)i . (31)Correlation functions for ˜ φ and ˜ χ can be obtained from the generating functional Z = Z D ˜ χ D ˜ φ ( Y n D C n D D n D A n D B n ) exp " − Z d x ˜ L + ∞ X n =1 L n + L int + J ˜ χ ˜ χ + J ˜ φ ˜ φ ! . (32)We Taylor-expand the exponential in (32) to second order in L int and integrate out all fieldsexcept ˜ φ and ˜ χ . This is equivalent to using the non-local Lagrangian12 (cid:16) ( ∂ i ˜ φ ( x )) + m φ, ˜ φ ( x ) + ( ∂ i ˜ χ ( x )) + m χ, ( x ) ˜ χ (cid:17) + ˜ g ˜ φ ( x ) ˜ χ ( x ) + Z d y (cid:16) ˜ φ ( x ) O ˜ φ ( x − y ) ˜ φ ( y ) + ˜ χ ( x ) O ˜ χ ( x − y ) ˜ χ ( y ) (cid:17) (33)with O ˜ φ ( x − y ) = 14 ˜ g ∞ X n =1 C n ( x ) C n ( x ) C n ( y ) C n ( y ) , (34)where we have used that the contributions from C n and D n are equal, and analogously O ˜ χ ( x − y ) = ˜ g ∞ X n =1 A n ( x ) C n ( x ) A n ( y ) C n ( y ) . (35)16o obtain a local Lagrangian we can Fourier transform O ˜ φ and O ˜ χ and expand in momentum k .We keep only the k = 0 term. This is equivalent to using the effective Lagrangian L eff = 12 ( ∂ i ˜ φ ) + 12 ˜ m φ ˜ φ + 12 ( ∂ i ˜ χ ) + 12 ˜ m χ ˜ χ + ˜ g ˜ φ ˜ χ . (36)Here ˜ m φ = m φ, + ˜ g π ) d X n d d p µ d − (cid:0) p + m χ, + ω n (cid:1) − = m φ, + ˜ g µ d − (4 π ) d/ Γ(2 − d )Γ(2) ∞ X n =1 ( m χ, + ω n ) d/ − (37)We have used dimensional regularisation; though each term in the sum over n is finite, the sumdiverges. The origin is the usual UV-divergence in the four dimensional theory. If we neglect themasses m φ, , m χ, in the loop we can perform the sum,˜ m φ = m φ, + ˜ g µ d − (4 π ) d/ Γ(2 − d )Γ(2) (2 πT ) d − ζ (4 − d ) . (38)Setting d = 3 − ǫ and expanding in ǫ we obtain˜ m φ = m φ, + g π ) (cid:18) ǫ + γ E + 12 [ ψ (1 / − log π ] + log h µT i(cid:19) (39)where γ E ≃ . ψ is the digamma function. Similarly we obtain˜ m χ = m χ, + g (4 π ) (cid:18) ǫ + γ E + 12 [ ψ (1 / − log π ] + log h µT i(cid:19) (40)The factor 2 difference in the loop correction can be identified with the symmetry factor of the φ -self-energy diagram. We can easily relate the temperature independent terms in (40) to theone-loop self-energy correction for vanishing external momentum in the four dimensional theory(1) in vacuum, g (4 π ) (cid:18) ǫ + γ E − log(4 π ) + log (cid:20) m χ µ (cid:21)(cid:19) (41)We impose renormalisation conditions at T = 0 and eliminate the logarithm in (41) by fixing fix µ = m , leading to a finite temperature contribution g (4 π ) log h m χ T i , (42)to ˜ m χ . There is an analogue correction g π ) log h m φ T i (43)to ˜ m φ . This shows that indeed the mass correction from heavy modes (hard thermal loops)is small, and we can confirm that hard modes can be treated perturbatively for our choice of17arameters. To estimate the effect of the light modes onto themselves, let us now consider a loopcorrection within the effective theory. 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